Planck scale boundary conditions in the standard model with singlet scalar dark matter
aa r X i v : . [ h e p - ph ] A p r SU-HET-06-2013IPMU13-0232
Planck scale boundary conditions in the standard modelwith singlet scalar dark matter
Naoyuki Haba , Kunio Kaneta and Ryo Takahashi Graduate School of Science and Engineering, Shimane University,Matsue, Shimane 690-8504, Japan Kavli IPMU (WPI), The University of Tokyo,Kashiwa, Chiba 277-8568, Japan
Abstract
We investigate Planck scale boundary conditions on the Higgs sectorof the standard model with a gauge singlet scalar dark matter. We willfind that vanishing self-coupling and Veltman condition at the Planckscale are realized with the 126 GeV Higgs mass and top pole mass, 172GeV . M t . . m S . Introduction
The Higgs particle has just been discovered at the Large Hadron Collider (LHC) experiment [1, 2].In addition, the results from the experiment are consistent with the standard model (SM), andan evidence of new physics such as supersymmetry (SUSY) is not obtained. Currently, theexperimental results strongly constrain the presence of SUSY at low energy although the minimalsupersymmetric standard model (MSSM) is an attractive candidate for new physics beyond theSM. Thus, a question, “How large is new physics scale?”, might become important for the SMand new physics. One can consider several scenarios such as high scale supersymmetric modelsor a scenario without SUSY in which the SM is valid up to the Planck scale M pl , etc.As an example of the later scenario, it was pointed out that imposing a constraint that theSM Higgs potential has two degenerate vacua, in which one of them is at the Planck scale,leads to the top mass 173 ± ± λ ( M pl ) = 0) and its β -function ( β λ ( M pl ) = 0) are imposed at the Planck scale. Inaddition to these two BCs, the work [5] also discussed the Veltman condition (Str M ( M pl )=0)and the vanishing anomalous dimension of the Higgs mass ( γ m h ( M pl ) = 0) at the Planck scale.It was found that the four BCs yield a Higgs mass range of 127 −
142 GeV. Thus, combiningthese BCs can interestingly predict values of the Higgs and top masses in the SM close to theexperimental ones but a slightly heavier Higgs mass and/or lighter top mass than experimentalones are generally predicted from these BCs as shown in [6] (see also [7, 8, 9, 10, 11, 12, 13] for thelatest analyses). BCs of λ ( M pl ) = 0 and Str M ( M pl )=0 mean that there exists an approximatelyflat direction in the Higgs potential, which might be adopted to the Higgs inflation [14, 15, 16,17, 18, 19, 20, 21, 22, 23]. In addition, the quadratic (logarithmic) divergence for the Higgs massdisappear at the Planck scale under the Veltman condition (the vanishing anomalous dimension γ m h ( M pl ) = 0). In this work, we investigate the three BCs in a gauge singlet extension of theSM.One important motivation for the gauge singlet extension of the SM is that the SM doesnot include a dark matter (DM). In the extension, the gauge singlet scalar can be DM whenthe scalar has odd parity under an additional Z symmetry [24] (see also [25, 26, 27]). Oncea gauge singlet scalar is added to the SM, an additional positive contribution from new scalarcoupling appears in the β -function of the Higgs self-coupling and the Veltman condition (andthe anomalous dimension of the Higgs mass). This means that the discussion of the three BCsat the Planck scale is modified from the SM. Since it actually seems difficult to reproduce 126 In order to confirm the existence of the flat direction, one should know the full ultraviolet (UV) completion.In the work, we assume an UV theory yielding the BCs for the flat direction at the Planck scale. See [28, 29] for discussions of the vacuum stability and triviality in the SM with a gauge singlet real scalar.See also [30] and references therein for implications of the LHC data to models with an extra singlet scalar, [31, 32]for the classically conformal U (1) B − L extended SM, [33] for a model of electroweak and conformal symmetrybreaking. . ± .
24 GeV top pole mass [34] (see also [35, 36]) at the same time(i.e. a slightly heavier Higgs and/or a lighter top masses than the experimental center values arerequired) under the above three BCs at the Planck scale in the SM, it is interesting to investigateif the BCs could be realized with the center values of the Higgs and top masses in the singletscalar DM extension of the SM. In this work, we take the following setup: (i) We consider asimple framework, in which only one gauge singlet real scalar is added to the SM. (ii) The gaugesinglet scalar is DM. (iii) All scalar quartic couplings in the model can be perturbatively treatedup to the Planck scale.The paper is organized as follows: In Section 2, we investigate the three BCs at the Planckscale in the above framework. As a result, we will find that the vanishing self-coupling andVeltman condition at the Planck scale are realized with the 126 GeV Higgs mass and top polemass, 171.8 GeV . M t . . m S . We consider the SM with a gauge singlet real scalar S , and investigate the values of scalar quarticcouplings at the Planck scale by solving renormalization group equations (RGEs) in the model.The relevant Lagrangian of the model and the RGEs for the scalar quartic couplings are givenby L = L SM + L S , (1) L SM ⊃ − λ (cid:18) | H | − v (cid:19) , (2) L S = − ¯ m S S − k | H | S − λ S S + (kinetic term) , (3)and (4 π ) dXdt = β X ( X = λ, k, λ S ) , (4)with β λ = µ < m H λ + 12 λy − y − λ ( g ′ + 3 g ) + [2 g + ( g ′ + g ) ] for m H ≤ µ < m S λ + 12 λy − y − λ ( g ′ + 3 g ) + [2 g + ( g ′ + g ) ] + k for m S ≤ µ , (5) β k = (cid:26) µ < m S k (cid:2) k + 12 λ + 6 y − ( g ′ + 3 g ) + λ S (cid:3) for m S ≤ µ , (6) β λ S = (cid:26) µ < m S λ S + 12 k for m S ≤ µ , (7)2espectively, where we assume that the Higgs mass m H is smaller than DM mass m S . H is theSM Higgs doublet, v is the vacuum expectation value (VEV) of the Higgs as 246 GeV, y is thetop Yukawa coupling, t is defined as t ≡ ln µ , and µ is a renormalization scale within the rangeof M Z ≤ µ ≤ M pl . We also impose an additional Z symmetry on the model. Only the gaugesinglet scalar has odd parity while all the SM fields have even parity under the symmetry. Wegive some comments on properties of the three scalar quartic couplings obeying Eqs. (4) ∼ (7): • The right-hand side of Eq. (6) is proportional to k itself. Thus, if we take a small valueof k ( M Z ), where M Z is the Z boson mass, a change of value in the running of k ( µ ) is alsosmall and remained in a small value. As a result, the running of λ closes to that of theSM. • It is known as the vacuum instability that the value of λ becomes negative before thePlanck scale in the SM with the experimental center values of the Higgs and top masses.This is due to the negative contribution from the top Yukawa coupling to the β -functionof λ as in Eq. (5). The minimum in the running of λ is around O (10 ) GeV. It is alsoshown from NNLO computations [6] that λ can remain positive up to the Planck scalewhen 127 GeV . m h .
130 GeV for M t = 173 . ± . . . M t . . m h = 126 GeV). • Once the gauge singlet scalar is added to the SM, the additional contribution of k / β -function of λ . This contribution can lift the running of λ ,and thus, λ can be around zero at the Planck scale. • The position of the minimum in the running of λ comes to lower energy scale than O (10 )GeV by adding the gauge singlet scalar because the contribution of k / • The realization of the vanishing λ around the Planck scale by adding the gauge singletscalar means that λ becomes negative before the Planck scale due to the above third andfourth properties of λ . Then, λ returns to zero. • The running of λ S is an increasing function of t (or µ ). There is not a direct contributionfrom λ S to the β -function of λ but the running of λ S affects on that of λ through therunning of k .We investigate the case that the gauge singlet scalar is DM with the three BCs ( λ ( M pl ) = 0, β λ ( M pl ) = 0, and Str M ( M pl ) = 0) in this model. Since we impose the odd-parity on thesinglet scalar under the additional Z symmetry, the singlet can be a candidate for DM. Thus,Ω S h = 0 .
119 must be reproduced in the case, where Ω S is the density parameter of the singletand h is the Hubble parameter. If m S < m H , β λ is zero for µ < m H and is given by the third line of right-hand side of Eq. (5) for m H ≤ µ . .1 Vanishing Higgs self-coupling: λ ( M pl ) = 0 First, we consider the BC that λ is zero at the Planck scale M pl = 10 GeV, λ ( M pl ) = 0. TheBCs of the Higgs self-coupling and top Yukawa coupling at low energy are taken as λ ( M Z ) = m h v = 0 . , y ( M t ) = √ m t ( M t ) v , (8)for the RGEs, where m h = 126 GeV is taken, M t is the top pole mass as 173 . ± .
24 GeV, and m t is the MS mass as 160 +5 − GeV [34]. Let us solve the RGEs, Eqs. (4) ∼ (7). Gray dots in Fig. 1 (a) show the region satisfying | λ ( M pl ) | < − and Ω S h = 0 . m S ≡ p ¯ m S + kv / . ± .
24 GeV are also depicted by the horizontal dashed lines. Figure 1 (b) is a typicalexample of the runnings of the scalar quartic couplings satisfying the above conditions (and theVeltman condition discussed later). The horizontal and vertical axes are the renormalizationscale and the values of scalar quartic couplings, respectively. Black, blue, and red curves indicatethe runnings of λ , k , and λ S , respectively. Initial conditions for the corresponding RGEs are k ( M Z ) = 0 . λ S ( M Z ) = 0 . M t = 173 GeV, and m S = 800 GeV with Eq. (8).It can be seen from Fig. 1 (a) that | λ ( M pl ) | < − can be satisfied in a region of 85 GeV . m S . . × GeV with the corresponding top pole mass, 171 . . M t . . m S & GeV. This is due to the following reason: A larger top massneeds a larger value of k in order to realize the tiny value of λ at the Planck scale. And a larger k requires a larger DM mass to give the correct abundance in the range of m S & GeV (e.g.,see [29, 37]).In order to realize the correct abundance of DM in m S . GeV, k ( M Z ) . . k ( M pl ) is not also large ( k ( M pl ) <
1) for the realization of DM. Since we have also imposedthe condition of 0 < λ S ( M pl ) < λ S does not actually affect onthe abundance of DM. Thus, the region described by the gray dots in Fig. 1 (a) is not changedeven with the condition of λ S ( M pl ) > λ ( M pl ) = 0, which isincluded in gray dots of Fig. 1 (a), but it is not possible. Since λ ( µ ) < GeV . µ < M pl )and λ ( M pl ) = 0, there is a global minimum of the potential between the EW and Planck scales.If one identifies the Higgs with the inflaton, the inflaton rolls downslope to the global minimumnot to the EW one. Thus, one must consider the other inflation models. We also take the following values as [34], sin θ W ( M Z ) = 0 . α − = 128, α s ( M Z ) = 0 .
118 for theparameters in the EW theory, where θ W is the Weinberg angle, α em is the fine structure constant, and α s is thestrong coupling, respectively. If one considers λ ( µ ) > λ ( M pl ), one would have a successful Higgs inflation with a non-minimal coupling of the Higgs field to the Ricci curvature scalar. λ ( M pl ) predicting close values of the Higgs and top masses to experimentalones at low energy motivates one to investigate the BC of λ ( M pl ) = 0 and/or the possibilityof the Higgs inflation. On the other hand, the value of λ S does not strongly affect the SM(Higgs and top masses), DM sectors, and other cosmology compared to that of λ (and k whichdetermines the abundance of DM). Thus, we focus only on the BC of λ ( M pl ) = 0 in this work.If there could be phenomenological and/or cosmological motivations to impose λ S ( M pl ) = 0, thediscussions of the realization of the BC might also be interesting. M ( M pl ) = 0 The Veltman condition, which indicates a disappearance of the quadratic divergence on the1-loop radiative correction to the bare Higgs mass, is modified toStr M v ≡ λ + k g ′ + 94 g − y = 0 , (9)where the term k/ k | H | S / k/ − .
291 when one takes m h = 126 GeV and M t = 173 .
07 GeV.We also show the region satisfying | Str M ( M pl ) /v | < − and Ω S h = 0 .
119 for the SMwith the singlet DM in Fig. 1 (a) by deep and light red dots. The deep and light red dots indicate10 − < λ S ( M Z ) < . . < λ S ( M Z ) <
1, respectively. One can see that | Str M ( M pl ) /v | < − can be satisfied in a region of 180 GeV . m S . . . M t . . λ and the Veltman condition can be satisfied in the region of300 GeV . m S . ,
172 GeV . M t . . . (10)We will return to this point later. This DM mass region will also be checked by the futureXENON100 experiment with 20 times sensitivity of the current data [37]. One can also see thata larger top mass requires a larger DM mass. The reason is similar to the case of the vanishing λ condition, i.e. a larger top mass needs a larger value of k in order to cancel the negativecontribution from − y term at the Planck scale, and thus a larger k requires a larger DM massto give the correct abundance in the range of m S & GeV.We also comment on the anomalous dimension for the Higgs mass defined by(4 π ) dm h dt = γ m h , (11)which indicates the logarithmic divergence. It is also modified to γ m h = m h (cid:18) λ + 6 y − g − g ′ (cid:19) + 2 km S , (12)5a) (b)
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D < Λ S <
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D - < Λ S < < Λ S < < Λ S <
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D - < Λ S H M Z L < < Λ S H M Z L < W S h = < Λ S H M pl L < < k H M pl L < M t = M t = - < Λ S < < Λ S < < Λ S <
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D Str M H M pl Λ H M pl L ~
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D È Γ m h (cid:144) m h H M pl LÈ < - È Str M H M pl L(cid:144) v È < - È Λ H M pl LÈ < -
100 10 Μ @ GeV D Λ , k , Λ S ´ - W S h = H M Z L = Λ S H M Z L = M t =
173 GeV m S =
800 GeV Λ k Λ S ´ - Figure 1: (a) Regions satisfying the BCs at the Planck scale, in which conditions | λ ( M pl ) | < − , | Str M ( M pl ) /v | < − , and | γ m h /m h ( M pl ) | < − are depicted as gray, (deep and light) red,and (deep and light) blue, respectively. The deep and light red (blue) dots indicate 10 − <λ S ( M Z ) < . . < λ S ( M Z ) < | Str M ( M pl ) /v | < − ( | γ m h /m h ( M pl ) | < − ),respectively. (b) A typical example of runnings of λ , k , and λ S , whose initial conditions arespecified by k ( M Z ) = 0 . λ S ( M Z ) = 0 .
34. The parameter set of the figure (b) corresponds toa point of ( m S , M t ) = (800 GeV ,
173 GeV) in the figure (a).where the last term of the right-hand side of Eq. (12) is new contribution from the gauge singletscalar. The value of the anomalous dimension for the Higgs mass in the SM at the Planck scaleis ( γ SM m h /m h ) | µ = M pl ≃ − . | γ m h /m h ( M pl ) | < − and Ω S h = 0 .
119 at the same time. The deep and light blue dots indicate 10 − <λ S ( M Z ) < . . < λ S ( M Z ) <
1, respectively. One can see that | γ m h /m h ( M pl ) | < − canbe satisfied in a region of 200 GeV . m S .
300 GeV with the corresponding top pole mass,171 . . M t . . λ term in Eq. (12). Therefore, a larger top mass requires a larger value of k (equivalentlyto m S ). However, the magnitude of the decrease of the anomalous dimension by a larger topmass is smaller than those of λ and Str M /v because the sign of contribution from the topYukawa coupling only in the anomalous dimension is positive unlike the λ and Str M /v cases(see Eqs. (5), (9), and (12)). As a result, a top mass dependence of the anomalous dimension isweaker compared to those of the vanishing λ and the Veltman condition. It should be mentionedthat there is also a region in which two conditions of the vanishing λ and γ m h can be realized atthe same time. 6 .3 Vanishing beta-function of the self-coupling: β λ ( M pl ) = 0 In the SM, the β -function of λ becomes tiny at the Planck scale. The value is about β SM λ ( M pl ) ≃ . × − when one takes the Higgs and top pole masses as m h = 126 GeV and M t = 173 . m S = 800 GeV, k ( M Z ) = 0 .
24, and λ S ( M Z ) = 0 .
34 in additionto m h = 126 GeV and M t = 173 .
07 GeV as an example in the context of the SM with the gaugesinglet field, the corresponding values of the β -function at the Planck scale become β λ ( M pl ) ≃ . × − . Therefore, the vanishing β -function of λ at the Planck scale in the SM with thegauge singlet cannot be satisfied because the runnings of λ is increasing from a negative valuedue to the effect of the singlet field as shown in Fig. 1 (b). In this extension of the SM, β λ ( µ )becomes zero at µ ∼ O (10 − GeV) not the Planck scale.According the above analyses, the BC of β λ ( M pl ) = 0 cannot be realized but two BCs of λ ( M pl ) = Str M ( M pl ) = 0 can be satisfied in the model. Since the result might indicate thatall the Higgs potential is induced from a quantum correction under the current circumstances,one has no warrant for β λ ( M pl ) = 0. Thus, the non-vanishing β -function can be compatiblyunderstood. Furthermore, there are also two additional β -functions ( β k and β λ S ) in this model.Since values of β k ( M pl ) and β λ S ( M pl ) cannot be zero when we impose λ ( M pl ) = 0 and the correctabundance of DM, the vanishing condition for only β λ ( M pl ) might be meaningless. Thus, in thiswork we take a stance of giving up the vanishing β -function at the Planck scale to predict theHiggs and top masses, although β λ ( M pl ) = λ ( M pl ) = 0 condition adopted in [3] predicted thevalues of the Higgs and top masses roughly close to experimental magnitudes. It is remarkable that there is a region, given in Eq. (10), satisfying two independent BCs atthe Planck scale ( λ ( M pl ) ≃ M ( M pl ) ≃ γ m h ( M pl ) ≃ λ ( M pl ) ≃ M ( M pl ) ≃
0, with the correct DM abundance is just a non-trivial result.The double coincidence means that the Higgs potential becomes flat at the Planck scale. Thegauge singlet scalar plays a crucial role for the realization, and it becomes DM with the correctabundance in the universe at present. The double coincidence with DM might be an alternativeprinciple to “multiple point criticality principle” discussed in Ref. [3], where a condition thatthe SM Higgs potential has two degenerate vacua was imposed. In the above analyses, we have limited the values of k ( M pl ) and λ S ( M pl ) to be less than 1.But, when one allows the values up to 4 π , the two regions for the Veltman condition and thevanishing anomalous dimension are changed. We also weaken the conditions of ( | Str M ( M pl ) /v | , One vacuum we live is the EW scale, and another one is the Planck scale. Under the condition, the vanishing λ and β λ are required.
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D < Λ S <
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D - < Λ S < < Λ S < < Λ S <
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D - < Λ S H M Z L < < Λ S H M Z L < < Λ S H M Z L < W S h = < Λ S H M pl L < Π < k H M pl L < Π M t = M t = - < Λ S < < Λ S < < Λ S <
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D Str M H M pl Λ H M pl L ~
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D È Γ m h (cid:144) m h H M pl LÈ < - È Str M H M pl L(cid:144) v È < - È Λ H M pl LÈ < -
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D < Λ S <
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D - < Λ S < < Λ S < < Λ S <
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D - < Λ S H M Z L < < Λ S H M Z L < < Λ S H M Z L < W S h = < Λ S H M pl L < Π < k H M pl L < Π M t = M t = - < Λ S < < Λ S < < Λ S <
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D Str M H M pl Λ H M pl L ~
100 200 500 1000 2000 5000171.5172.0172.5173.0173.5174.0174.5175.0 m S @ GeV D M t @ G e V D È Γ m h (cid:144) m h H M pl LÈ < È Str M H M pl L(cid:144) v È < È Λ H M pl LÈ < - Figure 2: (a) Regions satisfying the BCs with ( k ( M pl ) , λ S ( M pl )) < π , in which conditions( | λ ( M pl ) | , | Str M ( M pl ) /v | , | γ m h /m h ( M pl ) | ) < − are depicted as gray, red, and blue dots,respectively. The deep, light, and the lightest red (blue) dots indicate 10 − < λ S ( M Z ) < . . < λ S ( M Z ) <
1, 1 < λ S ( M Z ) < | Str M ( M pl ) /v | < − ( | γ m h /m h ( M pl ) | < − ),respectively. (b) Regions satisfying the BCs with ( k ( M pl ) , λ S ( M pl )) < π , in which conditions( | λ ( M pl ) | , | Str M ( M pl ) /v | , | γ m h /m h ( M pl ) | ) < (10 − , . , .
05) are depicted as gray, red andblue dots, respectively. | γ m h /m h ( M pl ) | ) < − to < .
05, the allowed regions for the conditions grow wider. Figure 2shows the cases, and Fig. 2 (a) shows regions satisfying the BCs with ( k ( M pl ) , λ S ( M pl )) < π at the Planck scale, in which conditions | λ ( M pl ) | < − , | Str M ( M pl ) /v | < − , and | γ m h /m h ( M pl ) | < − are depicted as gray, red, and blue dots, respectively. The deep, light, andthe lightest red (blue) dots indicate 10 − < λ S ( M Z ) < .
1, 0 . < λ S ( M Z ) <
1, 1 < λ S ( M Z ) < | Str M ( M pl ) /v | < − ( | γ m h /m h ( M pl ) | < − ), respectively. One can see that the regionsatisfying | Str M ( M pl ) /v | < − and | γ m h /m h ( M pl ) | < − grow wider compared to thecase shown in Fig. 1 (a) when one allows the value of λ S ( M Z ) up to 2, which corresponds to λ S ( M pl ) < π . Such a relatively large λ S ( M Z ) can effectively increase the value of k enoughto cancel the negative contribution in the Veltman condition and anomalous dimension at thePlanck scale even when one takes a smaller k ( M Z ). In this case, the double coincidence of λ ( M pl ) ≃ M ( M pl ) ≃ γ m h ( M pl ) ≃
0) still occurs.Figure 2 (b) shows regions satisfying the weaker BCs, | λ ( M pl ) | < − , | Str M ( M pl ) /v | < .
05, and | γ m h /m h ( M pl ) | < .
05 with ( k ( M pl ) , λ S ( M pl )) < π . In the case, the allowed regionsbecome the widest among all cases we have investigated. As a result, the region satisfying threeBCs at the same time appears around150 GeV . m S .
300 GeV ,
172 GeV . M t . . . (13)This means that the triple coincidence for the three BCs occurs. The triple coincidence requiresthat the logarithmic divergence of the Higgs mass also disappear at the Planck scale instead ofallowing a fine-tuning between the bare Higgs mass and a quadratic correction.8 Summary and discussions
We have investigated Planck scale BCs on the Higgs sector in the SM with gauge singletscalar DM. The BCs are the vanishing Higgs self-coupling ( λ ( M pl ) = 0), the Veltman condi-tion (Str M ( M pl )=0) (and the vanishing anomalous dimension for the Higgs mass parameter, γ m h ( M pl ) = 0), and the vanishing β -function of the self-coupling ( β λ ( M pl ) = 0). If one imposesthe BCs in the SM, the Higgs and top masses are predicted to be close to the experimentalones. BCs of λ ( M pl ) = 0 and Str M ( M pl )=0 mean that there exists approximately flat directionin the Higgs potential. In addition, the quadratic (logarithmic) divergence for the Higgs massdisappears under the BC of the Veltman condition (and the vanishing anomalous dimension atthe Planck scale). However, it actually seems difficult to reproduce 126 GeV Higgs mass an173 . ± .
24 GeV top pole mass at the same time under the three BCs in the SM. Thus, wehave investigated these BCs in the context of the SM with the singlet real scalar.We have taken the setup that the singlet is DM and all scalar quartic coupling in the modelcan be perturbatively treated up to the Planck scale. And we have utilized the Higgs with 126GeV mass in the analyses. We could find that the vanishing self-coupling and Veltman conditionat the Planck scale can be remarkably realized with the 126 GeV Higgs mass and top pole mass,172 GeV . M t . . m S . k ( M pl ) , λ S ( M pl )) < π and ( | Str M ( M pl ) /v | , | γ m h /m h ( M pl ) | ) < .
05, the triple coincidence ( λ ( M pl ) ≃
0, Str M ( M pl ) ≃
0, and γ m h ( M pl ) ≃ λ ( M pl ) = 0 implies that our EW vacuum is false and the true vacuum is at a high energy scaleslightly smaller than the Planck scale like the SM with the center values of the Higgs and topmasses. We have checked that the quantum tunnelling probability p through out the historyof the universe, which is estimated by p ≃ V U H exp( − π / (3 | λ ( H ) | )) (e.g., see [38]), can bemuch smaller than 1, where V U = τ U , τ U is the age of the universe as τ U ≃ . | λ ( H ) | ≃ . × − with H ≃ × GeV for the true vacuum of our sample point of m H = 126 GeV, M t = 173 GeV, m S = 800 GeV, k ( M Z ) = 0 .
24, and λ S ( M Z ) = 0 . β λ ( M pl ) = 0 with λ ( M pl ) = 0, which were firstlyconsidered in [3], in this single extension of the SM. β λ ( µ ) cannot be zero at the Planck scale with λ ( M pl ) = 0 in the extension because there is an additional positive contribution from kS | H | interaction to β λ ( µ ). β λ ( µ ) becomes zero at µ ∼ O (10 − GeV) (not the Planck scale) with λ ( M pl ) ≃ λ ( M pl ) = β λ ( M pl ) = 0 in this singlet extension of SM, the BCs predict about 145 GeVHiggs mass and 175 GeV top pole mass at M Z scale, which are ruled out by experiments.Finally, we also comment on other issues such as the existence of the tiny neutrino mass and9he baryon asymmetry of the universe (BAU), which cannot be explained in the SM. One popularexplanation is given by adding heavy right-handed Majorana neutrinos into the SM. These areknown as the seesaw mechanism and the leptogenesis for generating the tiny neutrino mass andBAU, respectively. In this example of the extension, there exist additional contributions fromthe neutrino Yukawa couplings to β λ , Str M , and γ m h . If the magnitude of the neutrino Yukawacouplings is smaller than O (10 − ), which corresponds to the right-handed neutrino Majorananeutrino mass smaller than O (10 ) GeV, the contributions are negligible in the BCs like theYukawa couplings of the bottom quark and tau. On the other hand, if the neutrino Yukawacouplings are larger than O (0 . λ ( M pl ) = 0, a larger k ( M Z ) (equivalently a heavier DM mass) is required because of anegative contribution from the neutrino Yukawa coupling to β λ . Such a negative contributionmay well cancel other positive contributions in β λ such that β λ ( M pl ) = 0 can be realized atthe same time. A larger k ( M Z ) is needed also for the BC of Str M ( M pl ) = 0 because thecontribution from the neutrino Yukawa to Str M is negative. Finally, an effect of the neutrinoYukawa coupling for γ m h is relatively non-trivial compared to the other BCs because the positivecontributions from (top and neutrino) Yukawa couplings to γ m h compete with the negative onefrom 12 λ term, i.e. larger (positive) Yukawa couplings lead smaller (negative) value of λ at thePlanck scale. Thus, an accurate numerical analysis is required. Effects in the BCs from additionalparticles and their mass scales strongly depend on a model for generating the tiny neutrino massand BAU, e.g. adding right-handed neutrinos, but such a model dependent analysis of the BCswith explanations of the neutrino mass and BAU in addition to DM might also be interesting. Acknowledgement
The authors thank M. Holthausen for answering our question. This work is partially supportedby Scientific Grant by Ministry of Education and Science, Nos. 00293803, 20244028, 21244036,23340070, and by the SUHARA Memorial Foundation. The works of K.K. and R.T. are sup-ported by Research Fellowships of the Japan Society for the Promotion of Science for YoungScientists. The work is also supported by World Premier International Research Center Initiative(WPI Initiative), MEXT, Japan.
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